1. CMB 2016 (vol 59 pp. 652)
 Su, Huadong

On the Diameter of Unitary Cayley Graphs of Rings
The unitary Cayley graph of a ring $R$, denoted
$\Gamma(R)$, is the simple graph
defined on all elements of $R$, and where two vertices $x$ and
$y$
are adjacent if and only if $xy$ is a unit in $R$. The largest
distance between all pairs of vertices of a graph $G$ is called
the
diameter of $G$, and is denoted by ${\rm diam}(G)$. It is proved
that for each integer $n\geq1$, there exists a ring $R$ such
that
${\rm diam}(\Gamma(R))=n$. We also show that ${\rm
diam}(\Gamma(R))\in \{1,2,3,\infty\}$ for a ring $R$ with $R/J(R)$
selfinjective and classify all those rings with ${\rm
diam}(\Gamma(R))=1$, 2, 3 and $\infty$, respectively.
Keywords:unitary Cayley graph, diameter, $k$good, unit sum number, selfinjective ring Categories:05C25, 16U60, 05C12 

2. CMB 2016 (vol 59 pp. 617)
 Nakashima, Norihiro; Terao, Hiroaki; Tsujie, Shuhei

Canonical Systems of Basic Invariants for Unitary Reflection Groups
It has been known that there exists a canonical system for every
finite real reflection group. The first and the third authors
obtained
an explicit formula for a canonical system in the previous paper.
In this article, we first define canonical systems for the finite
unitary reflection groups, and then prove their existence.
Our proof does not depend on the classification of unitary reflection
groups.
Furthermore, we give an explicit formula for a canonical system
for every unitary reflection group.
Keywords:basic invariant, invariant theory, finite unitary reflection group Categories:13A50, 20F55 

3. CMB 2013 (vol 57 pp. 254)
 Christensen, Ole; Kim, Hong Oh; Kim, Rae Young

On Parseval Wavelet Frames with Two or Three Generators via the Unitary Extension Principle
The unitary extension principle (UEP) by Ron and Shen yields a
sufficient condition for the construction of Parseval wavelet frames with
multiple generators. In this paper we characterize the UEPtype wavelet systems that
can be extended to a Parseval wavelet frame by adding just one UEPtype wavelet
system. We derive a condition that is necessary for the extension of a UEPtype
wavelet system to any Parseval wavelet frame with any number of generators, and
prove that this condition is also sufficient to ensure that an extension
with just two generators is possible.
Keywords:Bessel sequences, frames, extension of wavelet Bessel system to tight frame, wavelet systems, unitary extension principle Categories:42C15, 42C40 

4. CMB 2012 (vol 57 pp. 25)
 Bourin, JeanChristophe; Harada, Tetsuo; Lee, EunYoung

Subadditivity Inequalities for Compact Operators
Some subadditivity inequalities for matrices and concave functions also hold for Hilbert space operators, but (unfortunately!) with an additional $\varepsilon$ term. It seems not possible to erase this residual term. However, in case of compact operators we show that the $\varepsilon$ term is unnecessary. Further, these inequalities are strict in a certain sense when some natural assumptions are satisfied. The discussion also stresses on matrices and their compressions and several open questions or conjectures are considered, both in the matrix and operator settings.
Keywords:concave or convex function, Hilbert space, unitary orbits, compact operators, compressions, matrix inequalities Categories:47A63, 15A45 

5. CMB 2011 (vol 55 pp. 297)
 Glasner, Eli

The Group $\operatorname{Aut}(\mu)$ is Roelcke Precompact
Following a similar result of Uspenskij on the unitary group of a
separable Hilbert space, we show that, with respect to the lower (or
Roelcke) uniform structure, the Polish group $G=
\operatorname{Aut}(\mu)$ of automorphisms of an atomless standard
Borel probability space $(X,\mu)$ is precompact. We identify the
corresponding compactification as the space of Markov operators on
$L_2(\mu)$ and deduce that the algebra of right and left uniformly
continuous functions, the algebra of weakly almost periodic functions,
and the algebra of Hilbert functions on $G$, i.e., functions on
$G$ arising from unitary representations, all coincide. Again
following Uspenskij, we also conclude that $G$ is totally minimal.
Keywords:Roelcke precompact, unitary group, measure preserving transformations, Markov operators, weakly almost periodic functions Categories:54H11, 22A05, 37B05, 54H20 

6. CMB 2005 (vol 48 pp. 340)
 Andruchow, Esteban

Short Geodesics of Unitaries in the $L^2$ Metric
Let $\M$ be a type II$_1$ von Neumann algebra, $\tau$ a trace in $\M$,
and $\l2$ the GNS Hilbert space of $\tau$. We regard the unitary group
$U_\M$ as a subset of $\l2$ and characterize the shortest smooth
curves joining two fixed unitaries in the $L^2$ metric. As a
consequence of this we obtain that $U_\M$, though a complete (metric)
topological group, is not an embedded riemannian submanifold of $\l2$
Keywords:unitary group, short geodesics, infinite dimensional riemannian manifolds. Categories:46L51, 58B10, 58B25 

7. CMB 2003 (vol 46 pp. 54)
 Cheung, WaiShun; Li, ChiKwong

Linear Maps Transforming the Unitary Group
Let $U(n)$ be the group of $n\times n$ unitary matrices. We show that if
$\phi$ is a linear transformation sending $U(n)$ into $U(m)$, then $m$ is
a multiple of $n$, and $\phi$ has the form
$$
A \mapsto V[(A\otimes I_s)\oplus (A^t \otimes I_{r})]W
$$
for some $V, W \in U(m)$. From this result, one easily deduces the
characterization of linear operators that map $U(n)$ into itself obtained
by Marcus. Further generalization of the main theorem is also discussed.
Keywords:linear map, unitary group, general linear group Category:15A04 
